Last summer, Neymar opted to leave Barcelona for the Parc des Princes but just months into his long-term deal, it was claimed that the Brazilian was unhappy with life in France's capital.
Real have long been linked with an approach for the attacker and according to The Sun, the Champions League winners are ready to bid £307m in order to tempt PSG into a sale.
The report suggests that come September 1, a clause in Neymar's contract will allow the 26-year-old to leave for £263m, but it appears that Real are looking to accelerate any deal for an extra £44m.
Any approach would put the future of Cristiano Ronaldo in doubt, with the Spanish giants unlikely to be able to have both players on the wage bill after such a high outlay.
Earlier this week, it was claimed that Ronaldo would 'definitely' leave Real before the end of the summer window.